# Math Help - Connected Space

1. ## Connected Space

Hello, Can anyone give some clue on how I can determine if $R^{Z^+}$ is or isn't connected in the metric topology with distance
$\rho(x,y)=Sup\{\overline{d}(x_i,y_i):i\in Z^+\}$ where
$\overline{d}(a,b)=Min\{|a-b|,1\}$

2. Look at the bounded and unbounded sequences.

3. I think all sequenses are bounded because $\rho(x,y)\leq 1$ for all x,y

4. Not bounded in that sense. The space $\mathbb{R}^{\mathbb{Z}^+}$ is the space of sequences of real numbers, and such a sequence is bounded iff there's some real number M larger than the absolute value of every element of the sequence. Show that in the given topology, the set of bounded (respectively unbounded) sequences is open.